Civil Engineering Reference
In-Depth Information
Several types of stiffness may be defi ned, depending on the nature of applied loads. Structures
designed for vertical (gravity) loads generally possess suffi cient vertical stiffness. Earthquakes generate
inertial forces due to vibration of masses. Horizontal components of these inertial forces are often
dominant; hence lateral (or horizontal) stiffness is of primary importance for structural earthquake
engineers. The defi nition of the lateral stiffness, especially the secant value
K
s
, depends signifi cantly
on the region of interest in the response domain, i.e. the behaviour limit state of interest. The stiffness
of a system is associated primarily with satisfaction of the functionality (or serviceability) of the struc-
ture under dynamic loads. High deformability (and hence low stiffness) drastically reduces the structural
functionality.
In seismic design, adequate lateral stiffness is an essential requirement to control deformations,
prevent instability (local and global), prevent damage of non-structural components and ensure human
comfort during minor-to-moderate earthquakes. Human response to earthquakes is generally different
from the discomfort induced by other environmental actions, e.g. strong winds (Mileti and Nigg, 1984 ;
Durkin, 1985 ; Taranath, 1998). The reason is twofold. Firstly, earthquakes are less frequent than wind-
storms and have shorter duration; few seconds versus some minutes. Secondly, earthquakes may have
serious psychological effects, such as trauma, on people.
Lateral stiffness is infl uenced by properties of construction material, section type, members, connec-
tions and systems, which are linked hierarchically as shown in Figure 2.4. Further discussion is given
below.
2.3.1.1 Factors Infl uencing Stiffness
(i) Material Properties
Material properties that infl uence the structural stiffness are the elastic Young's modulus
E
and the
elastic shear modulus
G
. In the inelastic range, the lateral stiffness depends still on the moduli
E
and
G
, not on initial, but rather tangent values. The material stiffness is often evaluated through the ratio
of the elastic modulus
E
to the weight
γ
. Values of
E
/
γ
are 20 - 30 × 1 0
4
m for masonry and 200 - 300 ×
10
4
m for metals as also outlined in Table 2.3 . The specifi c elasticity
E
/
γ
of concrete is about 100- 150
× 1 0
4
m. Construction materials with low values of
E
/
γ
lead to stiff structures, e.g. masonry buildings
are stiffer than steel.
(ii) Section Properties
Section properties that affect the structural stiffness are the cross- sectional area
A
, the fl exural moment
of inertia
I
and the torsional moment of inertia
J
. Section area and fl exural inertia primarily infl uence
the axial, bending and shear stiffness of the system, for metal structures area (
A
) and moment of inertia
(
I
and
J
) do not change with types and levels of applied loads. Conversely, for masonry and RC, the
above properties are a function of the loading and boundary conditions. For example, the fl exural
moment of inertia
I
of RC rectangular members about the strong axis can be defi ned as shown in Figure
2.10; similarly, for the defi nition of the area
A
of RC cross sections. For elements in tension, it is gener-
ally assumed that only the steel reinforcement bars are effective because of the low tensile strength of
concrete.
The stiffness of the section is signifi cantly affected by modifi cations of its geometry. Figure 2.11
shows the variation of area
A
and fl exural moment of inertia about the strong axis
I
obtained by
increasing the size of beam and column members. In the fi gure, the subscript 1 refers to the original
section, while the subscript 2 is for the new section (original and added component). The dimensionless
results plotted in Figure 2.11 demonstrate that the increase in the inertia
I
is higher than the area
A
.
The results emphasize that by jacketing members, the previous balance between axial, torsional and
fl exural stiffness, and strength, is disturbed, hence a full reassessment of the original design is
warranted.
